# Clip 3/11: Lesson - Part 2

## Overview

In this lesson students think about the data "For 30 beans, Joe took 20 seconds to count, Sarah took 24 seconds to count, and Alex took 15 seconds to count." Students are asked to write a rate for each student and justify who is fastest. One student uses seconds per 30 beans, but most think about 30 beans per some amount of seconds. A girl observes that if the rate is seconds per bean it needs to be the smallest rate to win, but if it is beans per second it needs to be the largest rate.

COACH LINDA FISHER: Students are starting to go beyond answers and make generalizations about using rates to solve problems or make generalizations. They are starting to see that they compare whether one of the quantities is the same by looking at the differences in the other quantities. One girl was able to think that the rates could be written with either measure as the numerator, but then the results would be interpreted differently. This kind of thinking can't happen from working practice problems on a worksheet. The mathematics only arises because of context. So this idea from the pre-lesson about developing flexible thinking means not only being able to set up the rate either way, but also means stepping back and being able to interpret the two different ways of writing the ratio based on the meaning of the labels or units. This dimensional analysis seems pretty sophisticated for sixth graders.